A157461 Expansion of x*(x+1) / (x^2-26*x+1).

1, 27, 701, 18199, 472473, 12266099, 318446101, 8267332527, 214632199601, 5572169857099, 144661784084973, 3755634216352199, 97501827841072201, 2531291889651525027, 65716087303098578501, 1706086977990911515999, 44292545340460600837473

Offset

1,2

Comments

This sequence is part of a solution of a more general problem involving two equations, three sequences a(n), b(n), c(n) and a constant A:

A * c(n)+1 = a(n)^2,

(A+1) * c(n)+1 = b(n)^2, for details see comment in A157014.

A157461 is the b(n) sequence for A=6.

Numbers k such that 42*k^2 + 7 is a square. - Klaus Purath, Jun 12 2021

Links

Colin Barker, Table of n, a(n) for n = 1..700

Index entries for linear recurrences with constant coefficients, signature (26,-1).

Formula

G.f.: x*(x+1) / (x^2-26*x+1).

a(1) = 1, a(2) = 27, a(n) = 26*a(n-1)-a(n-2) for n>2.

a(n) = (13+2*sqrt(42))^(-n)*(-6-sqrt(42)+(-6+sqrt(42))*(13+2*sqrt(42))^(2*n))/12. - Colin Barker, Jul 25 2016

a(n+1) = (a(n)^2 - 28)/a(n-1), n > 1. - Klaus Purath, Jun 12 2021

Prog

(PARI) Vec(x*(x+1)/(x^2-26*x+1)+O(x^20)) \\ Charles R Greathouse IV, Sep 26 2012
(PARI) a(n) = round((13+2*sqrt(42))^(-n)*(-6-sqrt(42)+(-6+sqrt(42))*(13+2*sqrt(42))^(2*n))/12) \\ Colin Barker, Jul 25 2016

Crossrefs

6*A157874(n)+1 = A153111(n)^2.

7*A157874(n)+1 = A157461(n)^2.

Sequence in context: A113364 A095898 A014914 * A342037 A162827 A163179

Adjacent sequences: A157458 A157459 A157460 * A157462 A157463 A157464

Keyword

nonn,easy

Author

Paul Weisenhorn, Mar 01 2009

Extensions

Edited by Alois P. Heinz, Sep 09 2011

Status

approved